Mathematical Sciences: Geometry and Representation Theory

Kumar将继续他关于广义系数等变上同调的工作。他将研究G-空间的一个新的G-等变上同调理论，其中G是一个实李群。这个新的上同调理论被定义为复数的上同调，它是由李代数上的广义函数由李代数上的多项式函数得到的。这种上同调在研究G-流形上的横切椭圆算子的指标理论时遇到。李群理论是以挪威数学家索菲斯·李的名字命名的，一直是20世纪数学的主要主题之一。作为利用系统固有对称性的数学工具，李群的表示理论对数学本身，特别是在分析和数论方面，以及对理论物理，特别是量子力学和基本粒子物理，都产生了深远的影响。

Kumar will continue his work on equivariant cohomology with generalized coefficients. He will study a new G-equivariant cohomology theory of a G-space, where G is a real Lie group. This new cohomology theory is defined as the cohomology of the complex which is obtained from the polynomial functions on the Lie algebras of G by generalized functions on the Lie algebra. This cohomology is encountered in studying the index theory of transversally elliptic operators on G-manifolds. The theory of Lie groups, named in honor of the Norwegian mathematician Sophus Lie, has been one of the major themes in twentieth century mathematics. As the mathematical vehicle for exploiting the symmetries inherent in a system, the representation theory of Lie groups has had a profound impact upon mathematics itself, particularly in analysis and number theory, and upon theoretical physics, especially quantum mechanics and elementary particle physics.